We calculate asymptotic forms for the expected number of distinct sites, 〈${\mathit{S}}_{\mathit{N}}$(n)〉, visited by N noninteracting n-step symmetric L\'evy flights in one dimension. By a L\'evy flight we mean one in which the probability of making a step of j sites is proportional to 1/|j${\mathrm{|}}^{1+\mathrm{\ensuremath{\alpha}}}$ in the limit j\ensuremath{\rightarrow}\ensuremath{\infty}. All values of \ensuremath{\alpha}\ensuremath{\gtrsim}0 are considered. In our analysis each L\'evy flight is initially at the origin and both N and n are assumed to be large. Different asymptotic results are obtained for different ranges in \ensuremath{\alpha}. When n is fixed and N\ensuremath{\rightarrow}\ensuremath{\infty} we find that 〈${\mathit{S}}_{\mathit{N}}$(n)〉 is proportional to (${\mathit{Nn}}^{2}$${)}^{1/(1+\mathrm{\ensuremath{\alpha}})}$ for \ensuremath{\alpha}1 and to ${\mathit{N}}^{1/(1+\mathrm{\ensuremath{\alpha}})}$${\mathit{n}}^{1/\mathrm{\ensuremath{\alpha}}}$ for \ensuremath{\alpha}\ensuremath{\gtrsim}1. When \ensuremath{\alpha} exceeds 2 the second moment is finite and one expects the results of Larralde et al. [Phys. Rev. A 45, 7128 (1992)] to be valid. We give results for both fixed n and N\ensuremath{\rightarrow}\ensuremath{\infty} and N fixed and n\ensuremath{\rightarrow}\ensuremath{\infty}. In the second case the analysis leads to the behavior predicted by Larralde et al. \textcopyright{} 1996 The American Physical Society.